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\documentclass[journal=jctcce,manuscript=article]{achemso}

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\author{Christopher R. Sweet}
\altaffiliation{Center for Research Computing}
\affiliation[University of Notre Dame]
{Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, USA}
\author{Vijay S. Pande}
\affiliation[Stanford University]
{Department of Chemistry, Stanford University, Stanford, CA, USA}
\author{Jes\'{u}s A. Izaguirre}
\altaffiliation{Interdisciplinary Center for Network Science and Applications}
\email{izaguirr@nd.edu}
\affiliation[University of Notre Dame]
{Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, USA}

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\title[Flexible Block Method]
  {The Flexible Block Method for Computing Low-Frequency Normal Modes of Proteins}

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%\abbreviations{IR,NMR,UV}
%\keywords{American Chemical Society, \LaTeX}

\begin{document}
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\begin{abstract}
 We show an improvement upon the popular Rotation Translation Block
 (RTB) Method for computing low-frequency normal modes of
 macromolecules. RTB rests on the hypothesis that low-frequency normal
 modes can be described as pure rigid-body motions of blocks of
 consecutive amino-acid residues. We show that accuracy of the
 computed eigenvectors can be significantly improved by including
 internal flexibility of the amino-acid residue blocks, while
 maintaining the asymptotic complexity of RTB. Results on normal mode
 analyses of proteins ranging from 33 residues (WW domain) up to 858
 residues (dimeric citrate synthase) were performed to demonstrate
 these results. Theoretical connections to the Trace Minimization
 method are also shown.
\end{abstract}

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\section{Introduction}
The low frequency normal modes of proteins are of interest in
understanding their collective motions, which are often of
biological interest. The Rotation-Translation-Block method
\cite{Durand:1994vy} is one of the most popular and fast methods to
approximately compute these modes. 

Diagonalization methods such as those used in quantum chemistry
usually require a good initial approximations of the eigenvectors,
which has been done for proteins that have obvious hinge or other
collective motions~\cite{Dykeman:2010dh}. In
these cases, Lanczos methods can be profitably applied. Other
iterative methods have been proposed, although they usually converge
slowly~\cite{Kaledin:2006ug}. 

Many diagonalization methods use highly coarse-grained representations
of molecules to be computationally efficient. Chief among these are
the elastic network models, which have been extensively used to study
proteins and other
macromolecules~\cite{Atilgan01,DORUKER1,CRYB05,Eyal:2008jw,Bahar:2010gf}. Here,
we choose to work with all-atom models in Cartesian and dihedral
space, which are the most accurate.

FBM belongs to a class of methods that uses component modes and
coupling between them to compute approximations to the true low
frequency modes.  The component synthesis method (CSM) divides a
protein into 3 contiguous sections, computes the local normal modes of
each section and corrects these estimates by explicitly considering
interactions between neighbouring blocks~\cite{HAO92}. Component modes
are commonly used in the engineering literature as an intermediate
step in solving whole system eigenvalue
problems~\cite{Lee:2003wn,Chun:2000vo}. Other component based methods
are the Partial Hessian Vibrational Analysis, the Mobile Block
Hessian, and the Vibrational Subsystem
Analysis~\cite{Ghysels:2009de,Ghysels:2009wj,Ghysels:2009cp}.

FBM and RTB can also be seen as applications of an inexact Trace
Minimization
algorithm~\cite{Wisniewski:1982uy,Sameh:2000ba,Naumov:2008dz} where
the orthogonal trial matrices are formed from the residue-block model
but where there is no conjugate gradient solution at the end. This
viewpoint allows for rigorous error estimates and highlights potential
improvements.  

We will explore methods to guarantee convergence of FBM's calculations
to acceptable accuracy limits. In particular, we have started
analyzing perturbation-iteration methods~\cite{Durand:2000jz}, the
Mixed Basis method~\cite{Mouawad:1993we,Perahia:1995te}, the Component
Synthesis Method~\cite{HAO92} already described, and our own method
that uses iterations based on the low and high frequency components
left out from the block eigenvectors. 
\section{Methods}

\section{Numerical tests}

\section{Discussion}
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\begin{acknowledgement}


\end{acknowledgement}

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\bibliography{diagonalization}

\end{document}
